Some new results and inequalities for subsequences of Nörlund logarithmic means of Walsh–Fourier series

Abstract

We prove that there exists a martingale $f\in H_{p}$ such that the subsequence $\{L_{2^{n}}f \}$ of Nörlund logarithmic means with respect to the Walsh system are not bounded from the martingale Hardy spaces $H_{p}$ to the space $weak-L_{p}$ for $0< p<1$. We also prove that for any $f\in L_{p}$, $p\geq 1$, $L_{2^{n}}f$ converge to f at any Lebesgue point x. Moreover, some new related inequalities are derived.

1 Introduction

The terminology and notations used in this introduction can be found in Sect. 2.

It is well known that Vilenkin systems do not form bases in the space $L_{1}$. Moreover, there is a function in the Hardy space $H_{1}$, such that the partial sums of f are not bounded in the $L_{1}$-norm. Moreover, (see Tephnadze [22]) there exists a martingale $f\in H_{p}$ ($0< p<1$), such that

$$ \sup_{n\in \mathbb{N}} \Vert S_{2^{n}+1}f \Vert _{weak-L_{p}}=\infty . $$

On the other hand, (for details see, e.g., the books [20] and [25]) the subsequence $\{S_{2^{n}}\}$ of partial sums is bounded from the martingale Hardy space $H_{p}$ to the space $H_{p}$, for all $p>0$, that is, the following inequality holds:

$$ \Vert S_{2^{n}}f \Vert _{H_{p}}\leq c_{p} \Vert f \Vert _{H_{p}},\quad n\in \mathbb{N}, p>0. $$

(1)

It is also well known that (see [20] and [16])

$$ S_{2^{n}}f(x)\to f(x), \quad \text{for all Lebesgue points of } f \in L_{p}, \text{where } p\geq 1. $$

(2)

Weisz [26] considered the norm convergence of Fejér means of Vilenkin–Fourier series and proved that the inequality

$$ \Vert \sigma _{k}f \Vert _{p}\leq c_{p} \Vert f \Vert _{H_{p}},\quad p>1/2\text{ and }f\in H_{p}, $$

(3)

holds. Moreover, Goginava [8] (see also [12–15, 18]) proved that the assumption $p>1/2$ in (3) is essential. In particular, he showed that there exists a martingale $f\in H_{1/2}$ such that $\sup_{n\in \mathbb{N}} \Vert \sigma _{n}f \Vert _{1/2}=+ \infty$. However, Weisz [26] (see also [17]) proved that for every $f\in H_{p}$, there exists an absolute constant $c_{p}$, such that the following inequality holds:

$$ \Vert \sigma _{2^{n}}f \Vert _{H_{p}}\leq c_{p} \Vert f \Vert _{H_{p}},\quad n\in \mathbb{N}, p>0. $$

(4)

Móricz and Siddiqi [11] investigated the approximation properties of some special Nörlund means of Walsh–Fourier series of $L_{p}$ functions in norm. Approximation properties for general summability methods can be found in [2, 3]. Fridli, Manchanda and Siddiqi [5] improved and extended the results of Móricz and Siddiqi [11] to martingale Hardy spaces. The case when $\{ q_{k}=1/k:k\in \mathbb{N} \}$ was excluded, since the methods are not applicable to Nörlund logarithmic means. In [6] Gát and Goginava proved some convergence and divergence properties of the Nörlund logarithmic means of functions in the Lebesgue space $L_{1}$. In particular, they proved that there exists a function in the space $L_{1}$, such that $\sup_{n\in \mathbb{N}} \Vert L_{n}f \Vert _{1}=\infty$.

In [4] (see also [10]) it was proved that there exists a martingale $f\in H_{p}$, ($0< p< 1$) such that $\sup_{n\in \mathbb{N}} \Vert L_{n}f \Vert _{p}=\infty$.

In [19] (see also [24]) it was proved that there exists a martingale $f\in H_{1}$ such that

$$ \sup_{n\in \mathbb{N}} \Vert L_{n}f \Vert _{1}= \infty . $$

(5)

However, Goginava [7] proved that

$$ \Vert L_{2^{n}}f \Vert _{1}\leq c \Vert f \Vert _{1},\quad f\in L_{1}, n\in \mathbb{N}. $$

From this result it immediately follows that for every $f\in H_{1}$, there exists an absolute constant c, such that the inequality

$$ \Vert L_{2^{n}}f \Vert _{1}\leq c \Vert f \Vert _{H_{1}} $$

(6)

holds for all $n\in \mathbb{N}$. Goginava [7] also proved that for any $f\in L_{1}(G)$,

$$ L_{2^{n}}f(x)\to f(x),\quad \text{a.e., as } n\to \infty . $$

According to (1), (4) and (6), the following question is quite natural.

Question 1

Is the subsequence $\{L_{2^{n}} \}$ also bounded on the martingale Hardy spaces $H_{p}(G)$ when $0< p<1$?

In Theorem 2 of this paper we give a negative answer to this question. In particular, we further develop some methods considered in [1, 9] and prove that for any $0< p<1$, there exists a martingale $f\in H_{p}$ such that $\sup_{n\in \mathbb{N}} \Vert L_{2^{n}}f \Vert _{weak-L_{p}}= \infty$. Moreover, in our Theorem 1 we generalize the result of Goginava [7] and prove that for any $f\in L_{1}(G)$ and for any Lebesgue point x,

$$ L_{2^{n}}f(x)\to f(x),\quad \text{as } n\to \infty . $$

The main results in this paper are presented and proved in Sect. 4. Section 3 is used to present some auxiliary lemmas, where, in particular, Lemma 2 is new and of independent interest. In order not to disturb our discussions later some definitions and notations are given in Sect. 4. Finally, Sect. 5 is reserved for some open questions we hope can be a source of inspiration for further research in this interesting area.

2 Definitions and notations

Let $\mathbb{N}_{+}$ denote the set of the positive integers, $\mathbb{N}:=\mathbb{N}_{+}\cup \{0\}$. Denote by $Z_{2}$ the discrete cyclic group of order 2, that is $Z_{2}:=\{0,1\}$, where the group operation is the modulo 2 addition and every subset is open. The Haar measure on $Z_{2}$ is given so that the measure of a singleton is 1/2.

Define the group G as the complete direct product of the group $Z_{2}$, with the product of the discrete topologies of $Z_{2}$s. The elements of G are represented by sequences $x:=(x_{0},x_{1},\ldots,x_{j},\ldots)$, where $x_{k}=0\vee 1$.

It is easy to give a base for the neighborhood of $x\in G$, namely:

$$ I_{0} ( x ) :=G,\qquad I_{n}(x):=\{y\in G:y_{0}=x_{0}, \ldots,y_{n-1}=x_{n-1} \} \quad (n\in \mathbb{N}). $$

Denote $I_{n}:=I_{n} ( 0 )$, $\overline{I_{n}}:=G\backslash I_{n}$ and $e_{n}:= ( 0,\ldots,0,x_{n}=1,0,\ldots ) \in G$, for $n\in \mathbb{N}$. It is easy to show that $\overline{I_{M}}=\bigcup^{M-1}_{s=0}I_{s} \backslash I_{s+1}$.

If $n\in \mathbb{N}$, then every n can be uniquely expressed as $n=\sum_{k=0}^{\infty }n_{j}2^{j}$, where $n_{j}\in Z_{2}$ ($j\in \mathbb{N}$) and only a finite number of $n_{j}$ differ from zero. Let $\vert n \vert :=\max \{k\in \mathbb{N}: n_{k}\neq 0\}$.

The norms (or quasinorms) of the spaces $L_{p}(G)$ and $weak-L_{p} ( G )$, $( 0< p<\infty )$ are, respectively, defined by

$$ \Vert f \Vert _{p}^{p}:= \int _{G} \vert f \vert ^{p}\,d\mu ,\qquad \Vert f \Vert _{weak-L_{p}}^{p}:= \sup_{\lambda >0}\lambda ^{p}\mu\quad ( f>\lambda ) . $$

The kth Rademacher function is defined by

$$ r_{k} ( x ) := ( -1 ) ^{x_{k}} \quad( x\in G, k\in \mathbb{N} ) . $$

Now, define the Walsh system $w:=(w_{n}:n\in \mathbb{N})$ on G as:

$$ w_{n}(x):=\prod^{\infty }_{k=0}r_{k}^{n_{k}} ( x ) =r_{ \vert n \vert } ( x ) ( -1 ) ^{\sum _{k=0}^{ \vert n \vert -1}n_{k}x_{k}} \quad ( n\in \mathbb{N} ) . $$

It is well known that (see, e.g., [20])

$$\begin{aligned} w_{n} ( x+y ) =&w_{n} ( x )w_{n} ( y ). \end{aligned}$$

(7)

The Walsh system is orthonormal and complete in $L_{2} ( G )$ (see, e.g., [20]).

If $f\in L_{1} ( G )$ let us define Fourier coefficients, partial sums and the Dirichlet kernel by

$$\begin{aligned}& \widehat{f} ( k ) := \int _{G}fw_{k}\,d\mu\quad ( k \in \mathbb{N}\mathbbm{ } ) , \\& S_{n}f:=\sum_{k=0}^{n-1}\widehat{f} ( k ) w_{k},\qquad D_{n}:=\sum_{k=0}^{n-1}w_{k } \quad ( n\in \mathbb{N}_{+} ) . \end{aligned}$$

Recall that (for details see, e.g., [20]):

$$ D_{2^{n}} ( x ) =\textstyle\begin{cases} 2^{n}, & \text{if }x\in I_{n}, \\ 0, & \text{if } x\notin I_{n},\end{cases} $$

(8)

and

$$ D_{n}=w_{n}\sum^{\infty } _{k=0} n_{k}r_{k}D_{2^{k}}=w_{n}\sum ^{\infty } _{k=0} n_{k} ( D_{2^{k+1}}-D_{2^{k}} ),\quad \text{for }n=\sum ^{\infty }_{i=0} n_{i}2^{i}. $$

(9)

Let $\{ q_{k}:k\geq 0 \}$ be a sequence of nonnegative numbers. The Nörlund means for the Fourier series of f are defined by

$$ t_{n}f:=\frac{1}{{l_{n}}}\sum_{k=0}^{n}q_{n-k}S_{k}f. $$

In the special case when $\{q_{k}=1:k\in \mathbb{N}\}$, we obtain the Fejér means

$$ \sigma _{n}f:=\frac{1}{n}\sum_{k=1}^{n}S_{k}f . $$

If $q_{k}={1}/{(k+1)}$, then we obtain the Nörlund logarithmic means:

$$ L_{n}f:=\frac{1}{l_{n}}\sum_{k=0}^{n-1} \frac{S_{k}f}{n-k},\qquad l_{n}:= \sum_{k=1}^{n}\frac{1}{k}. $$

(10)

The Riesz logarithmic means are defined by

$$ R_{n}f:=\frac{1}{l_{n}}\sum_{k=1}^{n} \frac{S_{k}f}{k}, \qquad l_{n}:= \sum_{k=1}^{n}\frac{1}{k}. $$

We note that this is an inverse of the Nörlund logarithmic means.

The convolution of two functions $f,g\in L_{1}(G)$ is defined by

$$ ( f\ast g ) ( x ) := \int _{G}f ( x+t ) g ( t )\,d\mu (t)\quad ( x\in G ). $$

It is well known that if $f\in L_{p} ( G )$, $g\in L_{1} ( G )$ and $1\leq p<\infty$. Then, $f\ast g \in L_{p} ( G )$ and the corresponding inequality holds:

$$ \Vert f\ast g \Vert _{p}\leq \Vert f \Vert _{p} \Vert g \Vert _{1}. $$

(11)

The representations

$$ L_{n}f (x )=\int_{G}f (t )P_{n} (x+t )\,d\mu (t ) \quad \text{and}\quad R_{n}f (x )=\int_{G}f (t )Y_{n} (x+t )\,d\mu (t ) $$

for $n\in \mathbb{N}$ play a central role in the following, where

$$ P_{n}:=\frac{1}{Q_{n}}\sum^{n}_{k=1}q_{n-k}D_{k}\quad \text{and}\quad Y_{n}:=\frac{1}{Q_{n}} \sum ^{n}_{k=1} q_{k}D_{k} $$

are called the kernels of the Nörlund logaritmic and the Reisz means, respectively. It is well known that (see, e.g., Goginava [7] and Tephnadze [23]):

$$\begin{aligned} P_{2^{n}}(x)=D_{2^{n}}(x)-\psi _{2^{n}-1}(x){Y}_{2^{n}}(x). \end{aligned}$$

(12)

Moreover, for all $n\in \mathbb{N}$,

$$\begin{aligned} \Vert P_{2^{n}} \Vert _{1}< c< \infty \quad \text{and}\quad \Vert Y_{n} \Vert _{1}< c< \infty . \end{aligned}$$

(13)

In the case $f\in L_{1}(G)$ the maximal functions are given by

$$ M(f) (x)=\sup_{n\in \mathbb{N}} \frac{1}{ \vert I_{n} ( x ) \vert } \biggl\vert \int _{I_{n} ( x ) }f ( u )\,d\mu ( u ) \biggr\vert =\sup _{n\in \mathbb{N}} 2^{n} \biggl\vert \int _{I_{n} ( x ) }f ( u )\,d\mu ( u ) \biggr\vert . $$

It is well known (for details see, e.g., [20]) that if $f\in L_{1}(G)$, then

$$ \bigl\Vert M(f) \bigr\Vert _{\text{weak}-L_{1}}\leq \Vert f \Vert _{1}. $$

According to a density argument of Calderon–Zygmund (see [20]) we obtain that if $f\in L_{1} ( G )$, then

$$ 2^{n} \biggl\vert \int _{I_{n} ( x ) }f ( u )\,d\mu ( u ) \biggr\vert \to 0,\quad \text{as } n \to \infty . $$

A point x on the Walsh group is called a Lebesgue point of $f\in L_{1} ( G )$, if

$$ \lim_{n\rightarrow \infty }2^{n} \int _{I_{n}(x)}f (t )\,d\mu (t)=f ( x ) \quad \text{a.e. }x\in G.$$

According to (2) we find that if $f\in L_{1} ( G )$, then a.e. point is a Lebesgue point.

Let $f:= ( f^{ ( n ) },n\in \mathbb{N} )$ be a martingale with respect to $\digamma _{n} ( n\in \mathbb{N} )$, which are generated by the intervals $\{ I_{n} ( x ) :x\in G \}$ (for details see, e.g., [25]).

We say that a martingale belongs to Hardy martingale spaces $H_{p} ( G )$, where $0< p<\infty$ if $\Vert f \Vert _{H_{p}}:= \Vert f^{*} \Vert _{p}<\infty$, where $f^{\ast }:=\sup_{n\in \mathbb{N}} \vert f^{(n)} \vert$.

If $f= ( f^{ ( n ) }, n\in \mathbb{N} )$ is a martingale, then the Walsh–Fourier coefficients must be defined in a slightly different manner:

$$ \widehat{f} ( i ) :=\lim_{k\rightarrow \infty } \int _{G}f^{ ( k ) } ( x )w_{i} ( x )\,d\mu ( x ) . $$

3 Auxiliary results

The Hardy martingale space $H_{p} ( G )$ has an atomic characterization (see Weisz [25, 26]).

Lemma 1

A martingale $f= ( f^{ ( n ) }, n\in \mathbb{N} )$ is in $H_{p}$ ($0< p\leq 1$) if and only if there exist a sequence $( a_{k},k\in \mathbb{N} )$ of p-atoms, which means that they satisfy the conditions

$$ \int _{I}a_{k}\,d\mu =0, \qquad \Vert a_{k} \Vert _{\infty }\leq \mu ( I ) ^{-1/p} ,\qquad \operatorname{supp} ( a_{k} ) \subset I, $$

and a sequence $( \mu _{k},k\in \mathbb{N} )$ of real numbers such that for every $n\in \mathbb{N}$:

$$ \sum _{k=0}^{\infty }\mu _{k}S_{2^{n}}a_{k}=f^{ ( n ) },\quad \textit{where } \sum_{k=0}^{\infty } \vert \mu _{k} \vert ^{p}< \infty . $$

(14)

Moreover, $\Vert f \Vert _{H_{p}}\backsim \inf ( \sum_{k=0}^{ \infty } \vert \mu _{k} \vert ^{p} ) ^{1/p}$, where the infimum is taken over all decompositions of f of the form (14).

We also state and prove a new lemma of independent interest:

Lemma 2

Let $n\in \mathbb{N}$ and $x\in I_{2}(e_{0}+e_{1})\in I_{0}\backslash I_{1}$. Then,

$$\begin{aligned} \Biggl\vert \sum_{j=2^{2\alpha _{k}}}^{2^{2\alpha _{k}+1}} \frac{D_{j}}{2^{2\alpha _{k}+1}-j} \Biggr\vert \geq \frac{1}{3}. \end{aligned}$$

Proof

Let $x\in I_{2}(e_{0}+e_{1})\in I_{0}\backslash I_{1}$. According to (8) and (9) we obtain that

$$ D_{j} ( x ) =\textstyle\begin{cases} w_{j}, & \text{if }j \text{ is an odd number,} \\ 0, & \text{if } j \text{ is an even number,}\end{cases} $$

and

$$\begin{aligned} \sum_{j=2^{2\alpha _{k}}}^{2^{2\alpha _{k}+1}-1} \frac{D_{j}}{2^{2\alpha _{k}+1}-j}=\sum _{j=2^{2\alpha _{k}-1}}^{2^{2 \alpha _{k}}-1}\frac{w_{2j+1}}{2^{2\alpha _{k}+1}-2j-1}=w_{1} \sum_{j=2^{2\alpha _{k}-1}}^{2^{2 \alpha _{k}}-1}\frac{w_{2j}}{2^{2\alpha _{k}+1}-2j-1}. \end{aligned}$$

Since

$$\begin{aligned}& \sum_{j=2^{2\alpha _{k}-2}+1}^{2^{2\alpha _{k}-1}-1} \biggl\vert \frac{1}{ 2^{2\alpha _{k}+1}-4j+3}-\frac{1}{2^{2\alpha _{k}+1}-4j+1} \biggr\vert \\& \quad = \sum_{j=2^{2\alpha _{k}-2}+1}^{2^{2\alpha _{k}-1}-1} \frac{2}{(2^{2\alpha _{k}+1}-4j+3)(2^{2\alpha _{k}+1}-4j+1)} \\& \quad \leq \sum_{j=2^{2\alpha _{k}-2}+1}^{2^{2\alpha _{k}-1}-1} \frac{2}{(2^{2\alpha _{k}+1}-4j)(2^{2\alpha _{k}+1}-4j)} \\& \quad \leq \frac{1}{8}\sum_{j=2^{2\alpha _{k}-2}+1}^{2^{2\alpha _{k}-1}-1} \frac{1}{(2^{2\alpha _{k}-1}-j)(2^{2\alpha _{k}-1}-j)} \\& \quad \leq \frac{1}{8}\sum_{k=1}^{\infty } \frac{1}{k^{2}}\leq \frac{1}{8}+ \frac{1}{8}\sum _{k=2}^{\infty }\frac{1}{k^{2}} \\& \quad \leq \frac{1}{8}+ \frac{1}{8} \sum_{k=2}^{\infty } \biggl( \frac{1}{k-1}- \frac{1}{k} \biggr) \\& \quad \leq \frac{1}{8}+\frac{1}{8}=\frac{1}{4}, \end{aligned}$$

if we apply $w_{4k+2}=w_{2}w_{4k}=-w_{4k}$, for $x\in I_{2}(e_{0}+e_{1})$, we find that

$$\begin{aligned} \Biggl\vert \sum_{j=2^{2\alpha _{k}}}^{2^{2\alpha _{k}+1}-1} \frac{D_{j}}{2^{2\alpha _{k}+1}-j} \Biggr\vert =& \Biggl\vert w_{2^{2 \alpha _{k}+1}-2}+ \frac{w_{2^{2\alpha _{k}+1}-4}}{3}+\sum_{j=2^{2 \alpha _{k}-1}+1}^{2^{2\alpha _{k}}-1} \frac{w_{2j}}{2^{2\alpha _{k}+1}-2j-1} \Biggr\vert \\ =& \Biggl\vert \frac{w_{2^{2\alpha _{k}+1}-4}}{3}- w_{2^{2\alpha _{k}+1}-4}+ \sum _{j=2^{2\alpha _{k}-1}+1}^{2^{2\alpha _{k}}-1} \frac{w_{2j}}{2^{2\alpha _{k}+1}-2j-1} \Biggr\vert \\ =& \Biggl\vert \frac{2w_{2^{2\alpha _{k}+1}-4}}{3}+\sum_{j=2^{2 \alpha _{k}-2}+1}^{2^{2\alpha _{k}-1}} \biggl( \frac{w_{4j-4}}{2^{2\alpha _{k}+1}-4j+3}+ \frac{w_{4j-2}}{2^{2\alpha _{k}+1}-4j+1} \biggr) \Biggr\vert \\ =& \Biggl\vert \frac{2w_{2^{2\alpha _{k}+1}-4}}{3} +\sum_{j=2^{2 \alpha _{k}-2}+1}^{2^{2\alpha _{k}-1}} \biggl( \frac{w_{4j-4}}{2^{2\alpha _{k}+1}-4j+3}- \frac{w_{4j-4}}{2^{2\alpha _{k}+1}-4j+1} \biggr) \Biggr\vert \\ \geq & \frac{2}{3}-\sum_{j=2^{2\alpha _{k}-2}+1}^{2^{2\alpha _{k}-1}} \biggl\vert \frac{1}{2^{2\alpha _{k}+1}-4j+3}- \frac{1}{2^{2\alpha _{k}+1}-4j+1} \biggr\vert \\ \geq& \frac{2}{3}- \frac{1}{4}\geq \frac{1}{3}. \end{aligned}$$

The proof is complete. □

4 Main results

Our first main result reads:

Theorem 1

Let $p\geq 1$ and $f\in L_{p}(G)$. Then,

$$\begin{aligned} \Vert L_{2^{n}} f-f \Vert _{p} \to 0 \quad \textit{as } n\to \infty . \end{aligned}$$

(15)

Moreover, for all Lebesgue points of f,

$$ \lim _{n\rightarrow \infty }L_{2^{n}}f(x)=f(x). $$

Proof

Let $n\in \mathbb{N}$. By combining (11) and (13) we immediately obtain

$$ \Vert L_{2^{n}} f \Vert _{p} \leq c_{p} \Vert f \Vert _{p}\quad \text{for all } n\in \mathbb{N},$$

which immediately implies (15).

To prove a.e. convergence we use identity (12) to obtain that

$$\begin{aligned} L_{2^{n}}f (x )={}&\int _{G}f (t )P_{2^{n}} (x+t )=\int _{G} f (t )D_{2^{n}} (x+t )\,d\mu (t )\,d\mu (t ) \\ &{}-\int _{G} f (t )w_{2^{n}-1}(x+t){Y}_{2^{n}}(x+t)\,d\mu (t ):=I-II. \end{aligned}$$

By applying (2) we can conclude that $I=S_{2^{n}}f(x)\to f(x)$ for all Lebesgue points of $f\in L_{p}$. Moreover, by using (7) we find that

$$\begin{aligned} II=\psi _{2^{n}-1}(x)\int _{G} f (t ){Y}_{2^{n}}(x+t){ \psi }_{2^{n}-1}(t)\,d(t). \end{aligned}$$

In view of (13) we see that

$$ f (t ){Y}_{2^{n}}(x+t)\in L_{p} \quad \text{where } p \geq 1 \text{ for any } x\in G, $$

and also note that II describes the Fourier coefficients of an integrable function. Hence, according to the Riemann–Lebesgue Lemma it vanishes as $n\to \infty$, i.e., $II\to 0$ for any $x\in G$, $n\to \infty$.

The proof is complete. □

Our next main result is the following answer of Question 1.

Theorem 2

Let $0< p<1$. Then, there exists a martingale $f\in H_{p}$ such that

$$ \sup_{n\in \mathbb{N}} \Vert L_{2^{n}}f \Vert _{weak-L_{p}}= \infty . $$

Proof

Let $\{ \alpha _{k}:k\in \mathbb{N} \}$ be an increasing sequence of the positive integers such that

$$\begin{aligned}& \sum_{k=0}^{\infty }\alpha _{k}^{-p/2}< \infty , \end{aligned}$$

(16)

$$\begin{aligned}& \sum_{\eta =0}^{k-1} \frac{ ( 2^{2\alpha _{\eta }} ) ^{1/p}}{\sqrt{ \alpha _{\eta }}}< \frac{ ( 2^{2\alpha _{k}} ) ^{1/p}}{\sqrt{\alpha _{k}}}, \end{aligned}$$

(17)

and

$$ \frac{ ( 2^{2\alpha _{k-1}} ) ^{1/p}}{\sqrt{\alpha _{k-1}}}< \frac{2^{2\alpha _{k}-8}}{\alpha _{k}^{1/2}l_{2^{2\alpha _{k}+1}}}. $$

(18)

Let

$$ f^{ ( n ) } ( x ) :=\sum_{ \{ k; 2 \alpha _{k}< n \} }\lambda _{k}a_{k}, $$

where

$$ \lambda _{k}=\frac{1}{\sqrt{\alpha _{k}}}\quad \text{and}\quad a_{k}={2^{2\alpha _{k}(1/p-1)}} ( D_{2^{2\alpha _{k}+1}}-D_{2^{2 \alpha _{k}}} ) . $$

From (16) and Lemma 1 we find that $f \in H_{p}$. It is easy to show that

$$ \widehat{f}(j)=\textstyle\begin{cases} \frac{2^{2\alpha _{k}(1/p-1)}}{\sqrt{\alpha _{k}}},& \text{if }j\in \{ 2^{2\alpha _{k}},\ldots, 2^{2\alpha _{k}+1}-1 \} , k\in \mathbb{N}, \\ 0,& \text{if }j\notin \bigcup_{k=1}^{\infty } \{ 2^{2 \alpha _{k}},\ldots, 2^{2\alpha _{k}+1}-1 \} . \end{cases} $$

(19)

Moreover,

$$\begin{aligned} L_{2^{2\alpha _{k}+1}}f &=\frac{1}{l_{2^{2\alpha _{k}+1}}}\sum _{j=1}^{2^{2 \alpha _{k}}-1}\frac{S_{j}f}{2^{2\alpha _{k}+1}-j}+\frac{1}{l_{2^{2\alpha _{k}+1}}}\sum_{j=2^{2\alpha _{k}}}^{2^{2\alpha _{k}+1}-1} \frac{S_{j}f}{{2^{2\alpha _{k}+1}-j}} \\ &:=I+II. \end{aligned}$$

(20)

Let $j<2^{2\alpha _{k}}$. By combining (17), (18) and (19) we can conclude that

$$\begin{aligned} \bigl\vert S_{j}f ( x ) \bigr\vert \leq &\sum _{\eta =0}^{k-1} \sum_{v=2^{2\alpha _{\eta }}}^{2^{2\alpha _{\eta }+1}-1} \bigl\vert \widehat{f}(v) \bigr\vert \\ \leq& \sum_{\eta =0}^{k-1} \sum_{v=2^{2 \alpha _{\eta }}}^{2^{2\alpha _{\eta }+1}-1} \frac{2^{2\alpha _{\eta }(1/p-1)}}{\sqrt{\alpha _{\eta }}} \\ \leq &\sum_{\eta =0}^{k-1} \frac{2^{2\alpha _{\eta }/p}}{\sqrt{\alpha _{\eta }}} \\ \leq& \frac{2^{2\alpha _{k-1}/p+1}}{\sqrt{\alpha _{k-1}}}< \frac{2^{2\alpha _{k}-4}}{\alpha _{k}^{1/2}l_{2^{2\alpha _{k}+1}}}. \end{aligned}$$

Hence,

$$\begin{aligned} \vert I \vert \leq &\frac{1}{l_{2^{2\alpha _{k}+1}}} \sum _{j=1}^{2^{2\alpha _{k}}-1} \frac{ \vert S_{j}f ( x ) \vert }{2^{2\alpha _{k}+1}-j} \\ \leq &\frac{1}{l_{2^{2\alpha _{k}+1}}} \frac{2^{2\alpha _{k-1}/p}}{\sqrt{\alpha _{k-1}}}\sum_{j=1}^{M_{2\alpha _{k}+1}-1} \frac{1}{j} \\ \leq& \frac{2^{2\alpha _{k-1}/p}}{\sqrt{\alpha _{k-1}}}. \end{aligned}$$

(21)

Let $2^{2\alpha _{k}}\leq j\leq 2^{2\alpha _{k}+1}-1$. We can write that

$$\begin{aligned} S_{j}f =&\sum_{\eta =0}^{k-1}\sum _{v=2^{2\alpha _{\eta }}}^{2^{2 \alpha _{\eta }+1}-1}\widehat{f}(v)w_{v}+ \sum_{v=2^{2\alpha _{k}}}^{j-1} \widehat{f}(v)w_{v} \\ =&\sum_{\eta =0}^{k-1} \frac{2^{{2\alpha _{\eta }} ( 1/p-1 ) }}{\sqrt{\alpha _{\eta }}} ( D_{2^{2\alpha _{\eta }+1}}-D_{2^{2 \alpha _{\eta }}} ) + \frac{2^{{2\alpha _{k}} ( 1/p-1 ) }}{\sqrt{\alpha _{k}}} ( D_{j}-D_{2^{{2\alpha _{k}}}} ) . \end{aligned}$$

It follows that

$$\begin{aligned} II={}&\frac{1}{l_{2^{2\alpha _{k}+1}}} \sum _{j=2^{2\alpha _{k}}} ^{2^{2\alpha _{k}+1}} \frac{1}{2^{2\alpha _{k}+1}-j} \Biggl( \sum_{\eta =0}^{k-1} \frac{2^{2\alpha _{\eta } ( 1/p-1 ) }}{\sqrt{\alpha _{\eta }}} ( D_{2^{2\alpha _{\eta }+1}}-D_{2^{2\alpha _{\eta }}} ) \Biggr) \\ &{}+\frac{1}{l_{2^{2\alpha _{k}+1}}} \frac{2^{2\alpha _{k} ( 1/p-1 ) }}{\sqrt{\alpha _{k}}}\sum_{j=2^{2\alpha _{k}}}^{2^{2\alpha _{k}+1}-1} \frac{ (D_{j}-D_{2^{2\alpha _{k}}} )}{2^{2\alpha _{k}+1}-j} \\ :={}&II_{1}+II_{2}. \end{aligned}$$

(22)

Let $x\in I_{2}(e_{0}+e_{1})\in I_{0}\backslash I_{1}$. According to $\alpha _{0}\geq 1$ we obtain that $2\alpha _{k}\geq 2$, for all $k\in \mathbb{N}$ and if we use (8) we obtain that $D_{2^{2\alpha _{k}}}=0$,

$$ II_{1}=0 $$

(23)

and

$$\begin{aligned} II_{2} =&\frac{1}{l_{2^{2\alpha _{k}+1}}} \frac{2^{2\alpha _{k}(1/p-1)}}{\sqrt{\alpha _{k}}}\sum _{j=2^{2\alpha _{k}-1}+1}^{2^{2\alpha _{k}}-1} \frac{w_{2j+1}}{2^{2\alpha _{k}+1}-2j-1} \\ =&\frac{1}{l_{2^{2\alpha _{k}+1}}} \frac{2^{2\alpha _{k}(1/p-1)}w_{1}}{\sqrt{\alpha _{k}}}\sum_{j=2^{2\alpha _{k}-1}+1}^{2^{2\alpha _{k}}-1} \frac{w_{2j}}{2^{2\alpha _{k}+1}-2j-1}. \end{aligned}$$

By using Lemma 2 we can conclude that

$$ \vert II_{2} \vert \geq \frac{1}{3} \frac{1}{l_{2^{2\alpha _{k}+1}}} \frac{2^{2\alpha _{k}(1/p-1)}}{\sqrt{\alpha _{k}}}\geq \frac{1}{l_{2^{2\alpha _{k}+1}}}\frac{2^{2\alpha _{k}(1/p-1)-1}}{\sqrt{\alpha _{k}}}. $$

(24)

If we apply (18), (20)–(24) for $x\in I_{2}(e_{0}+e_{1})$ and $0< p<1$, we have that

$$\begin{aligned} \bigl\vert L_{2^{2\alpha _{k}+1}}f (x ) \bigr\vert \geq & II_{2}-II_{1}-I \\ \geq &\frac{1}{l_{2^{2\alpha _{k}+1}}} \frac{2^{2\alpha _{k}(1/p-1)-2}}{\sqrt{\alpha _{k}}}-\frac{1}{l_{2^{2\alpha _{k}+1}}} \frac{2^{2\alpha _{k}(1/p-1)-3}}{\sqrt{\alpha _{k}}} \\ \geq &\frac{1}{l_{2^{2\alpha _{k}+1}}} \frac{2^{2\alpha _{k}(1/p-1)-3}}{\sqrt{\alpha _{k}}} \\ \geq& \frac{2^{2\alpha _{k}(1/p-1)-3}}{ (\ln 2^{2\alpha _{k}+1}+1 )\sqrt{\alpha _{k}}} \\ \geq & \frac{2^{2\alpha _{k}(1/p-1)-3}}{( 4\alpha _{k}+1)\sqrt{\alpha _{k}}} \geq \frac{2^{2\alpha _{k}(1/p-1)-6}}{\alpha ^{3/2}_{k}}. \end{aligned}$$

Hence, we can conclude that

$$\begin{aligned}& \Vert L_{q_{\alpha _{k}}^{s}}f \Vert _{weak-L_{p}} \\& \quad \geq \frac{2^{2\alpha _{k}(1/p-1)-6}}{\alpha ^{3/2}_{k}}\mu \biggl\{ x\in G: \vert L_{2^{2\alpha _{k}+1}}f \vert \geq \frac{2^{2\alpha _{k}(1/p-1)-6}}{\alpha ^{3/2}_{k}} \biggr\} ^{1/p} \\& \quad \geq \frac{2^{2\alpha _{k}(1/p-1)-6}}{\alpha ^{3/2}_{k}}\mu \biggl\{ x\in I_{2}(e_{0}+e_{1}): \vert L_{2^{2\alpha _{k}+1}}f \vert \geq \frac{2^{2\alpha _{k}(1/p-1)-6}}{\alpha ^{3/2}_{k}} \biggr\} ^{1/p} \\& \quad \geq \frac{2^{2\alpha _{k}(1/p-1)-6}}{\alpha ^{3/2}_{k}}\bigl(\mu \bigl( I_{2}(e_{0}+e_{1}) \bigr) \bigr)^{1/p} \\& \quad > \frac{c2^{2\alpha _{k}(1/p-1)}}{\alpha _{k}^{3/2}}\rightarrow \infty , \quad \text{as }k\rightarrow \infty . \end{aligned}$$

The proof is complete. □

5 Open questions

It is known (for details see, e.g., the books [20] and [25]) that the subsequence $\{S_{2^{n}}\}$ of the partial sums is bounded from the martingale Hardy space $H_{p}$ to the Lebesgue space $L_{p}$, for all $p>0$. On the other hand, (see Tephnadze [22]) there exists a martingale $f\in H_{p}$ ($0< p<1$), such that $\sup _{n\in \mathbb{N}} \Vert S_{2^{n}+1}f \Vert _{weak-L_{p}}=\infty$. However, Simon [21] proved that for all $f\in H_{p}$, there exists an absolute constant $c_{p}$, depending only on p, such that

$$ \sum ^{\infty }_{k=1} \frac{ \Vert S_{k}f \Vert _{p}^{p}}{k^{2-p}}\leq c_{p} \Vert f \Vert _{H_{p}}^{p},\quad ( 0< p< 1 ). $$

In [24] it was proved that for all $f\in H_{p}$, there exists an absolute constant $c_{p}$, depending only on p, such that

$$ \sum ^{\infty }_{k=1} \frac{ \Vert L_{k}f \Vert _{p}^{p}}{k^{2-p}}\leq c_{p} \Vert f \Vert _{H_{p}}^{p}, \quad ( 0< p< 1 ). $$

Open Problem 1

(a) Let $f\in H_{p}$, where $0< p<1$. Does there exist an absolute constant $c_{p}$, such that the following inequality holds:

$$ \sum ^{\infty } _{k=1} \frac{\log ^{p} k \Vert L_{k}f \Vert _{p}^{p}}{k^{2-p}} \leq c_{p} \Vert f \Vert _{H_{p}}^{p},\quad ( 0< p< 1 )? $$

(b) For $0< p<1/2$ and any nondecreasing function $\Phi :\mathbb{N}\rightarrow {}[ 1,\infty )$ satisfying the conditions $\lim _{n\rightarrow \infty }\Phi ( n ) =+ \infty$, is it possible to find a martingale $f\in H_{p}$ such that

$$ \sum_{n=1}^{\infty } \frac{\log ^{p} n \Vert L_{n}f \Vert _{p}^{p}\Phi (n )}{n^{2-p}} =\infty ? $$

Open Problem 2

(a) Let $f\in H_{p}$, where $0< p\leq 1$ and

$$ \omega _{H_{p}} \biggl( \frac{1}{2^{n}},f \biggr) =o \biggl( \frac{\log n}{2^{n(1/p-1)}\log ^{2[p]}n} \biggr), \quad \text{as } n\rightarrow \infty . $$

Does the following convergence result hold:

$$ \Vert L_{k}f-f \Vert _{H_{p}}\rightarrow 0,\quad \text{as }k \rightarrow \infty ? $$

(b) Let $0< p\leq 1$. Does there exist a martingale $f\in H_{p}$, for which

$$ \omega _{H_{p}} \biggl( \frac{1}{2^{n}},f \biggr) =O \biggl( \frac{\log n}{2^{n(1/p-1)}\log ^{2[p]}n} \biggr) ,\quad \text{as }n \rightarrow \infty $$

and $\Vert L_{k}f-f \Vert _{weak-L_{p}}\nrightarrow 0$, as $k\rightarrow \infty$?